On Differential Systems with Quadratic Impulses and Their Applications to Lagrangian Mechanics

Abstract

This paper is concerned with the basic dynamics and a class of variational problems for control systems of the form \[({\text{E}})\qquad \dot x = f(t,x,u) + g(t,x,u)\dot u + h(t,x,u)\dot u^2 \] These systems have impulsive character, due to the presence of the time derivative $\dot u$ of the control. It is shown that trajectories can be well defined when the controls u are limits (in a suitable weak sense) of sequences $(u_n )$ contained in the Sobolev space $W^{1,2} $. Roughly speaking, one can say that, in this case, the $\dot u_n $ tend to the square root of a measure. Actually, this paper shows that the system (E) is essentially equivalent to an (affine) impulsive system of the form \[\dot x = f(x) + g(x)v + h(x)\dot w,\] where $v \in L^2 $ and $\dot w$ is a nonnegative Radon measure not smaller than $v^2 $. This provides a characterization of the closure of the set of trajectories of (E), as the controls u range inside a fixed ball of $W^{1,2} $. The existence of (generalized) optimal controls for variational problems of Mayer type is also investigated. Since the main motivation for studying systems of form (E) comes from Rational Mechanics, this paper concludes by presenting an example of an impulsive Lagrangian system.